Products of locally minimal groups

Abstract

A topological group G is called locally minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f:G→H is a continuous isomomorphism such that f(V) is a neighbourhood of the identity in H, then f is open. This paper is focused on the study of products of locally minimal groups. Local minimality is not preserved under taking products. We call a topological groups G perfectly locally minimal group if G×H is locally minimal for every locally minimal group H. The well known criterion of Stoyanov for perfectly minimal abelian groups is extended, first to the case of arbitrary perfectly minimal groups. Then its counterpart for perfectly locally minimal groups is provided. In particular, we show that a topological group G is perfectly locally minimal if and only if G×(Z,τp) is locally minimal for every prime p, where τp denotes the p-adic topology on Z. Complete locally minimal groups turn out to be perfectly locally minimal. This motivates a special attention towards (local) minimality of complete groups. A contribution is given towards Uspenksij's problem on minimality of complete groups, by proving that if {Gi:i∈I} is a family of minimal groups with minimal quotient Gi/Z(Gi) the product G=∏i∈IGi is minimal if and only if Z(G) is minimal. Furthermore, we answer a question from [14] about closed subgroups of a product of locally compact abelian groups, showing that they are locally compact if and only if they are locally minimal. Finally, a general criterion for local minimality of arbitrary products is given based on already known criteria for minimality of arbitrary products [10].